> While it has been claimed that Born's law can be derived from the many-worlds interpretation, the existing proofs have been criticized as circular.
So it seems you aren't alone in feeling this way.
Edit: Forgot the link - https://en.wikipedia.org/wiki/Born_rule
The latter article is more recent at 2005 but as far as I can tell Carroll's self-locating uncertainty ideas weren't introduced until around 2014.
The TL;DR is that the critics are correct. Deriving the Born rule begs the question because it makes an unjustified assumption (branching indifference) and also introduces an "invisible pink unicorn", i.e. a concept that, according to the theory, has physical significance but cannot be measured. In the case of MWI that concept is branch weights.
[UPDATE] The critique I wrote is based on Wallace (https://arxiv.org/abs/0906.2718). Carroll's argument appears to be somewhat different. I'm just now working my way through his paper (https://arxiv.org/abs/1405.7577) but I'd be very surprised if it did not also have some untenable assumption hidden in there somewhere.
I have previously come across Deutsch's formulation in terms of information flow , along with what seemed like a very strong criticism from Wallace and Christopher Timpson  that his model was not gauge-invariant.